2
Fuzzy numbers and fuzzy vectors
Taking care of the fuzziness of data described in Chapter it is necessary to have a mathematical model to describe such data in a quantitative way. This is the subject of Chapter 2.
2.1 Fuzzy numbers and characterizing functions
In order to model one-dimensional fuzzy data the best up-to-date mathematical model is so-called fuzzy numbers.
Definition 2.1: A fuzzy number x* is determined by its so-called characterizing function ξ(·) which is a real function of one real variable x obeying the following:
1. ξ : → [0; 1].
2. ∀δ ∈ (0; 1] the so-called δ-cut Cδ(x*) :={x ∈ : ξ(x) ≥ δ} is a finite union of compact intervals, .
3. The support of ξ(·), defined by supp[ξ(·)] :={x ∈ : ξ(x) > 0} is bounded.
The set of all fuzzy numbers is denoted by .
For the following and for applications it is important that characterizing functions can be reconstructed from the family (Cδ(x*); δ ∈ (0; 1]), in the way described in Lemma 2.1.
Lemma 2.1:
For the characterizing function ...